Notes on Contributors
Alisa Bokulich is Professor of Philosophy at Boston University and Director of the Center for Philosophy & History of Science, where she organizes the Boston Colloquium for Philosophy of Science. She is an Associate Member of Harvard University’s History of Science Department and Series Editor for Boston Studies in the Philosophy and History of Science. She is the author of Reexamining the QuantumClassical Relation: Beyond Reductionism and Pluralism (Cambridge University Press, 2008) and her research focuses on scientific models and explanations in the physical sciences, including the geosciences.
Mazviita Chirimuuta is Associate Professor of History and Philosophy of Science at the University of Pittsburgh. She received her PhD in visual neuroscience from the University of Cambridge in 2004, and held postdoctoral fellowships in philosophy at Monash University (2005–8) and Washington University St. Louis (2008–9). Her principal area of research is in the philosophy of neuroscience and perceptual psychology, and her book Outside Color: Perceptual Science and the Puzzle of Color in Philosophy was published by MIT Press in 2015.
Mark Colyvan is Professor of Philosophy at the University of Sydney and a Visiting Professor at the Munich Center for Mathematical Philosophy at the Ludwig-Maximilians University in Munich. He holds a BSc (Hons) in mathematics (from the University of New England) and a PhD in philosophy (from the Australian National University). His main research interests are in the philosophy of mathematics, philosophy of logic, decision theory, risk analysis, and philosophy of ecology and conservation biology. He is the author of The Indispensability of Mathematics (Oxford University Press, 2001), Ecological Orbits: How Planets Move and Populations Grow (Oxford University Press, 2004, with co-author Lev Ginzburg), An Introduction to the Philosophy of Mathematics (Cambridge University Press, 2012), and numerous papers.
John Cusbert is a Research Fellow in Philosophy at the University of Oxford. He has a PhD in philosophy from the Australian National University. His research focuses on various topics in and around probability and decision theory, ethics, and metaphysics.
Steven French is Professor of Philosophy of Science at the University of Leeds. He is Co-Editor in Chief of the British Journal for the Philosophy of Science and Editor in Chief of the Palgrave Macmillan series New Directions in Philosophy of Science. His most recent book is The Structure of the World: Metaphysics and Representation (Oxford University Press, 2014) and his next one is Applying Mathematics: Immersion, Inference, Interpretation (with Otavio Bueno).
Lina Jansson is Assistant Professor of Philosophy at the University of Nottingham. She received her PhD from the University of Michigan, Ann Arbor, and previously worked at Nanyang Technological University in Singapore. She works on issues related to explanation, laws of nature, and confirmation from within the history and philosophy of science, general philosophy of science, and the philosophy of physics. She has published work on issues related to non-causal explanation, explanations in Newton’s Principia, ground, parsimony, and probability in Everettian quantum theories.
Marc Lange is Theda Perdue Distinguished Professor and Philosophy Department Chair at the University of North Carolina at Chapel Hill. He is the author of Because Without Cause: Non-Causal Explanation in Science and Mathematics (Oxford University Press, 2016), Laws and Lawmakers (Oxford University Press, 2009), An Introduction to the Philosophy of Physics: Locality, Fields, Energy, and Mass (Blackwell, 2002), and Natural Laws in Scientific Practice (Oxford University Press, 2000).
Kelvin McQueen is Assistant Professor of Philosophy and affiliate of the Institute for Quantum Studies at Chapman University. He has a PhD in philosophy from the Australian National University. He works on a variety of topics in the philosophy of science, the philosophy of physics, the philosophy of mind, and metaphysics.
Margaret Morrison is Professor of Philosophy at the University of Toronto and is a fellow of the Royal Society of Canada and the Leopoldina-German National Academy of Sciences. She received her PhD in philosophy of science from the University of Western Ontario. Her work covers a broad range of topics in the philosophy of science including physics and biology. Some of her publications include Reconstructing Reality: Models, Mathematics and Simulation (Oxford University Press, 2015), Unifying Scientific Theories (Cambridge University Press, 2000), and over sixty articles in various journals and edited collections.
Christopher Pincock is Professor of Philosophy at the Ohio State University. He works on topics at the intersection of the philosophy of science and the philosophy of mathematics. He is the author of Mathematics and Scientific Representation (Oxford University Press, 2012).
Angela Potochnik is Associate Professor at the University of Cincinnati. She earned her PhD at Stanford University. She works on a variety of topics in philosophy of science, including methodological issues in population biology, especially evolutionary and behavioral ecology; idealized models in biology and in science more generally; scientific explanation; relations among different projects and fields of science; how gender and social factors influence science; and the history of logical empiricism, especially the work of Otto Neurath. She is the author of, Idealization and the Aims of Science (University of Chicago Press, 2017).
Alexander Reutlinger is Assistant Professor at the Ludwig-MaximiliansUniversität München (Munich Center for Mathematical Philosophy). He works on
topics in philosophy of science and neighbouring areas of epistemology and metaphysics (including topics such as explanation, causation, probabilities, ceteris paribus laws, idealizations, reduction, and models). He previously held positions as a postdoctoral research fellow at the University of Cologne and as a visiting fellow at the University of Pittsburgh’s Center for Philosophy of Science.
Juha Saatsi is Associate Professor at the University of Leeds. He works on various topics in philosophy of science, and he has particular interests in the philosophy of explanation and the scientific realism debate.
Michael Strevens is Professor of Philosophy at New York University. He has written on scientific explanation, complexity, probability and probabilistic inference, causation, the social structure of science, and concepts of natural kinds and other theoretical concepts. He previously taught at Stanford University and Iowa State University.
James Woodward is Distinguished Professor in the Department of History and Philosophy of Science at the University of Pittsburgh. Prior to 2010 he was the J. O. and Juliette Koepfli Professor at the California Institute of Technology. He is the author of Making Things Happen: A Theory of Causal Explanation (Oxford University Press, 2003) which won the 2005 Lakatos award, a past president of the Philosophy of Science Association (2010–12), and a fellow of the American Academy of Arts and Sciences.
Introduction Scientific Explanations Beyond Causation
Alexander Reutlinger and Juha Saatsi
What is a scientific explanation? This has been a central question in philosophy of science at least since Hempel and Oppenheim’s pivotal attempt at an answer in 1948 (also known as the covering-law model of explanation; Hempel 1965: chapter 10). It is no surprise that this question has retained its place at the heart of contemporary philosophy of science, given that it is one of the sciences’ key aims to provide explanations of phenomena in the social and natural world around us. As philosophers of science, we naturally want to grasp and to explicate what exactly scientists are doing and aiming to achieve when they explain something.
In his classic Four Decades of Scientific Explanation, Salmon (1989) details the shift from Hempel and Oppenheim’s “epoch-making” logical empiricist beginnings to a mixture of subsequent perspectives on scientific explanation involving ideas concerning causation, laws, theoretical unification, pragmatics, and statistics. Although Salmon believes that causal accounts of explanation (including his own version) are considerably successful, he ultimately advocated a pluralistic outlook. According to his pluralism, different approaches to explanation are worth pursuing and they should be understood as complementing one another rather than competing with each other. He articulates this pluralism, for instance, in his claim about the “peaceful coexistence” of causal and unificationist accounts.1 According to Salmon, the four decades of intense philosophical activity on scientific explanation since 1948 did not result in anything like a consensus, and his prediction was that no broad consensus was likely to emerge after 1989, at least not in the short term.
However, Salmon’s pluralist outlook and his portrayal of the history of the debate (articulated in his Four Decades) were largely lost in subsequent philosophical work. The two decades following the publication of Salmon’s book in 1989 became the
1 Salmon’s well-known illustration of his pluralism is captured in the story of the friendly physicist (Salmon 1989: 183).
decades of causal accounts of explanation. As causal accounts came to dominate the philosophical scene, this tendency also resulted in establishing a research focus on causation itself, and since the late 1980s philosophers have made considerable progress in analysing various aspects of causation. For example, they have explicated different notions of causation, causal processes, causal mechanisms, and causal models, and they have achieved a better understanding of the connection between causes and different kinds of idealizations, of the link between causation and temporal order, and, indeed, of the kinds of explanations that causal information supports. According to causal accounts, the sciences explain by identifying the causes of the phenomenon to be explained—or, according to the mechanist version of causal accounts, by identifying the causal mechanisms for that phenomenon (for surveys see Andersen 2014; Woodward 2014).
Causal accounts have been considered to be attractive for several reasons. The focus on causal-mechanical aspects of explanation has undoubtedly been in many ways a good response to the shortcomings of the covering-law model (and of some alternative approaches to explanation). Moreover, the proponents of causal accounts have also taken a closer look at detailed case studies of real-life explanations in the sciences instead of merely analysing toy examples. The proponents of causal accounts have also advanced the field by taking seriously case studies from the life and social sciences, freeing the debate from a (formerly) widespread physics chauvinism. And, indeed, many paradigmatic explanations in the sciences rely on information about causes and mechanisms. Hence, philosophers focusing on causal explanation have achieved a great deal by studying this aspect of the explanatory practices of science. As a result, today hardly anyone denies the explanatory significance and epistemic value of causalmechanistic information provided by the sciences.
The domination of the causal accounts has shaped the subsequent debate on scientific explanation in several respects: in how arguments have been perceived and evaluated; what the criteria for an adequate account of scientific explanation have been taken to be (for instance, everybody had to talk about flagpoles, for better or worse), and so on. This spirit of a ‘causal hegemony’ can easily be detected in extant survey papers (such as Woodward 2014; Craver and Tabery 2017),2 also in influential works advocating a causal approach to scientific explanation (for instance, Woodward 2003; Craver 2007; Strevens 2008), and last but certainly not least in the tacit presumptions and ‘common knowledge’ one encounters at various conferences and workshops.
The state of the field after six long decades suggests that something close to a consensus was reached: scientific explanation is a matter of providing suitable information about causes of the explanandum phenomenon. However, over the past decade or so this consensus has come under increasing scrutiny and suspicion as philosophers have more widely begun to rethink the hegemony of causal-mechanist accounts.
2 However, Woodward’s entry in the Stanford Encyclopedia of Philosophy remains open-minded about the possibility of non-causal explanations (Woodward 2014: §7.1).
There are important precedents to this recent development. Indeed, although causal accounts did indeed dominate the philosophical scene in the 1990s and the 2000s, they were far from being the only game in town. From early on, a number of authors have drawn attention to non-causal ways of explaining, in particular in relation to unificationist accounts (Friedman 1974; Kitcher 1984, 1989; Bartelborth 1996), pragmatic accounts (van Fraassen 1980, 1989; Achinstein 1983), analyses of asymptotic explanations in physics (Batterman 2000, 2002), statistical and geometrical explanations (Lipton 1991/2004; Nerlich 1979), and other specific examples from various scientific disciplines (for instance, Forge 1980, 1985; Sober 1983; Ruben 1990/2012; Frisch 1998; Hüttemann 2004).
Over the past few years, this resistance to the causal hegemony has burgeoned quickly, and the present volume demonstrates this turning of the tide. Looking at the current literature, one particularly striking recent development is the increasing interest in the limits of causal accounts of explanation. The guiding idea is that although causation is certainly part of the truth about scientific explanation, it is unlikely to be the full story. Following this idea, philosophers have begun to explore the hypothesis that explanations in science sometimes go beyond causation. For instance, there seem to be genuinely non-causal explanations whose explanatory resources go ‘beyond causation’ as these explanations do not work by way of truthfully representing the causes of the phenomenon to be explained. Other scientific explanations go ‘beyond causation’ in the sense that their explanatory assumptions do not tell us anything about the causal mechanisms involved. In this spirit, a number of philosophers have argued that the repertoire of explanatory strategies in the sciences is considerably richer than causal accounts suggest. (See Reutlinger 2017 for a detailed survey of the present debate on non-causal explanations.)
The motivation for this shift of focus to explanations that go ‘beyond causation’ is easy to appreciate: there are plenty of compelling, real-life examples of non-causal explanations that causal accounts of explanation seemingly fail to capture. To be more precise, the new development in the philosophy of scientific explanation is the increasing recognition of interesting and varied examples of non-causal explanations of empirical phenomena to be found across the natural and social sciences.
Unsurprisingly, physics is a fertile ground for such examples, ranging from explanations involving symmetries and inter-theoretic relations, to theoretically more abstract explanations that rely on, for instance, renormalization group techniques. Moreover, in the more fundamental domains of physical theorizing, it seems relatively easy to find explanations that seem non-causal—in the first blush at least. Perhaps this does not come as a surprise to those sympathetic to increasingly popular scepticism about causation as a fundamental metaphysical category in physics (originating in the work of Ernst Mach and Bertrand Russell among others; see, for instance, Mach 1905; Russell 1912/13; Scheibe 2007: chapter 7). Such causal ‘anti-foundationalism’ is a contested topic in its own right, of course, but perhaps the difficulty of interpreting fundamental physics in plain causal terms already indicates that explanations
in fundamental physics operate in terms that go beyond causation (Price 1996; Price and Corry 2007).
One need not plunge the depths of fundamental physics to find compelling instances of non-causal explanations, however. Various philosophers have suggested that there are other kinds of non-causal explanations in the life and social sciences, such as mathematical, statistical, computational, network, optimality, and equilibrium explanations. Moreover, some of the most popular examples in the philosophical literature—the present volume included—involve rather simple empirical set-ups of strawberries and bridge-crossings. Philosophers’ love of toy examples is due to the fact that simple though such examples are, they are sufficiently instructive to challenge the philosophy of explanation centred around causal accounts, giving rise to fruitful engagement between competing philosophical analyses. For instance, what explains the fact that 23 strawberries cannot be distributed equally among 3 philosophers (cf. Chapter 1)? Is this explanation non-causal? Is it non-causal because it is mathematical? Is it mathematical in some distinct kind of way (in which familiar mathematized, and possibly causal, explanations in science are not)? As the essays in this volume demonstrate, thinking carefully about some exceedingly simple cases alongside real-life scientific explanations is not only fun, but philosophically profitable!3
Let us pause for a second. Surely, one might think, the existence of non-causal explanations is old news. After all, the empirical sciences are not the only epistemic project striving for explanations. Proofs in logic and pure mathematics are at least sometimes taken to be explanatory—and if so, then proofs explain in a non-causal way (see, for instance, Mancosu 2015). In metaphysical debates, too, one finds a straightforward appeal to non-causal explanations: for instance, if some fact A grounds another fact B, then A is taken to be non-causally explanatory of B (see, for instance, Bliss and Trogdon 2016). However, the fact that mathematicians, logicians, and metaphysicians sometimes explain in non-causal terms is an interesting and related topic but it is not the crucial motivation for questioning the hegemony of causal-mechanist accounts of explanations in the natural and social sciences.4 But even if non-causal explanations in logic, mathematics, and metaphysics do not motivate a challenge to causal hegemony in philosophy of science, it is certainly worth exploring the relationship between non-causal explanations in mathematics, logic, and metaphysics, on the one hand, and non-causal explanations in the natural and social sciences, on the other hand.
3 Action or teleological explanations are also often treated as a particular kind of non-causal explanation, as, for instance, von Wright (1971, 1974) argues. However, the allegedly non-causal character of action explanations is (infamously) controversial and has led to an extensive debate (see Davidson 1980 for a defence of a causal account of action explanations). We will bracket the debate on action explanations in this volume.
4 Although the existence of non-causal explanations internal to, for instance, pure mathematics and logic has long been recognized, detailed philosophical accounts of such explanations have been under-developed. The dominance of causal models of explanation in philosophy of science is partly to be blamed, since much of this work did not seem to be applicable or extendible to domains such as mathematics, where the notion of causation obviously does not apply.
Now, what would be an appropriate philosophical reaction to examples of non-causal explanations from the natural and social sciences? Let us canvass in the abstract three possible ‘big picture’ reactions:
1. causal reductionism,
2. explanatory pluralism, and
3. explanatory monism.
First, while some are happy to give up the hegemony of causal accounts of explanation and to welcome non-causal ways of explaining empirical phenomena, others feel less pressure to do so. Some philosophers—including some featured in this volume—take the seeming examples of non-causal explanations to rather point to the need for a more sophisticated account of causal explanation. If the seemingly non-causal explanations can ultimately be understood as causal explanations after all, perhaps non-causal explanations of empirical phenomena are indeed rare and exotic (if not wholly nonexistent). The attraction of such causal reductionism about explanation, if indeed true, lies in the fundamental causal unity it finds underlying the prima facie disparate activity of scientific explanation. One and the same conceptual framework provides a pleasingly unified philosophical theory of explanation, if all explanations in science—including alleged examples of non-causal explanations—turn out to ultimately function by providing causal information. In other words, causal reductionists would like to maintain and to defend the hegemony of causal explanation (see, for instance, Lewis 1986; more recently Skow 2014, 2016).
Second, one way to deny such causal reductionism is to accept some kind of explanatory pluralism. Pluralists adopt, roughly put, the view that causal and noncausal explanations are different types of explanations that are covered by two (or more) distinct theories of explanation.5 The core idea of a pluralist response to the existence of examples of causal and non-causal explanations is that causal accounts of explanations have to be supplemented with further accounts of non-causal explanations (a view Salmon was attracted to, as pointed out above, see Salmon 1989; more recently Lange 2016).
Third, an alternative to explanatory pluralism is explanatory monism: the view that there is one single philosophical account capable of capturing both causal and noncausal explanations by virtue of some ‘common core’ that they share. To take an analogy, consider the way in which some theories of explanation (such as Hempel’s or Woodward’s) account for both deterministic and probabilistic (causal) explanations. In an analogous way, a monist holds that one theory of explanation may account for both causal and non-causal explanation. Unlike the causal reductionist, the monist does not deny the existence of non-causal explanations. Rather, a monist holds that causal and non-causal
5 This notion of explanatory pluralism has to be distinguished from another kind of pluralist (or relativist) attitude towards explanations, according to which one phenomenon has two (or more) explanations and these explanations are equally well suited for accounting for the phenomenon.
explanations share a feature that makes them explanatory (for a survey of different strategies to articulate monism, see Reutlinger 2017).
The ‘big picture’ issue emerging from these three reactions is whether causal reductionism, explanatory pluralism, or explanatory monism provides the best approach to thinking about the similarities and differences between various causal and (seemingly) non-causal explanations of empirical phenomena. However, this ‘big picture’ question is far from being the only one, and we predict that these debates are likely to continue in the foreseeable future due to a number of other outstanding questions such as the following ones:
• How can accounts of non-causal explanations overcome the problems troubling the covering-law model?
• What is the best way to distinguish between causal and non-causal explanations?
• Which different types of non-causal explanations can be found in the life and social sciences?
• Is it possible to extend accounts of non-causal explanation in the sciences to non-causal explanations in other ‘extra-scientific’ domains, such as metaphysics, pure mathematics, logic, and perhaps even to explanations in the moral domain?
• What should one make of the special connection that some non-causal explanations seem to bear to certain kinds of idealizations?
• What role does the pragmatics of explanation play in the non-causal case?
• What are the differences between non-causal and causal explanatory reasoning, from a psychological and epistemological perspective?
• What does scientific understanding amount to in the context of non-causal explanations?
Let us now turn to a preview of the volume, which divides into three parts.
Part I addresses issues regarding non-causal explanations from the perspective of general philosophy of science. By articulating suitable conceptual frameworks, and by drawing on examples from different scientific disciplines, the contributions to this part examine and discuss different notions of non-causal explanation and various philosophical accounts of explanation for capturing non-causal explanations.
Marc Lange presents a view that is part of a larger pluralist picture. For him, there is no general theory covering all non-causal explanations, let alone all causal and noncausal explanations taken together. But Lange argues that a broad class of non-causal explanations works by appealing to constraints, viz. modal facts involving a stronger degree of necessity than physical or causal laws. Lange offers an account of the order of explanatory priority in explanations by constraint, and uses it to distinguish different kinds of such explanations. He illustrates the account with paradigmatic examples drawn from the sciences.
Christopher Pincock probes different strategies for spelling out what pluralism— the view that, roughly put, explanations come in several distinct types—amounts to in relation to causal vs. non-causal explanations. He contrasts ontic vs. epistemic versions
of pluralism, and he finds room within both versions to make sense of explanatory pluralism in relation to three types of explanations: causal, abstract, and constitutive types of explanation. Moreover, he also draws attention to several problems that explanatory pluralism raises requiring further consideration and, thereby, setting a research agenda for philosophers working in a pluralist spirit.
Angela Potochnik argues that theories of explanation typically have a rather narrow focus on analysing explanatory dependence relations. However, Potochnik argues that there is no good reason for such a narrow focus, because there are many other features of explanatory practices that warrant philosophical attention, i.e., other features than the causal or non-causal nature of explanatory dependence relations. The purpose of Potochnik’s contribution is mainly to convey to the reader that it is a serious mistake to ignore these ‘other features’. She draws philosophical attention to features of explanations such as the connection between explanation and understanding, the psychology of explanation, the role of (levels of) representation for scientific explanation, and the connection between the aim of explanation and other aims of science. Her contribution is a plea for moving the debate beyond causal—and also beyond non-causal—dependence relations.
Alexander Reutlinger defends a monist approach to non-causal and causal explanations: the counterfactual theory of explanation. According to Reutlinger’s counterfactual theory, both causal and non-causal explanations are explanatory by virtue of revealing counterfactual dependencies between the explanandum and the explanans (illustrated by five examples of non-causal scientific explanations). Moreover, he provides a ‘Russellian’ strategy for distinguishing between causal and non-causal explanations within the framework of the counterfactual theory of explanation. Reutlinger bases this distinction on ‘Russellian’ criteria that are often associated with causal relations (including causal asymmetry, time asymmetry, and distinctness).
Michael Strevens proposes to resist the popular view that some explanations are non-causal by virtue of being mathematical explanations. To support his objection, Strevens provides a discussion of various explanations that other philosophers regard as instances of non-causal qua being mathematical explanations (such as equilibrium explanations and statistical explanations). He argues that, at least in the context of these examples, the mathematical component of an explanation helps scientists to get a better understanding of (or a better grasp on) the relevant causal components cited in the explanation. Hence, Strevens’s contribution could be read as defending a limited and careful version of causal reductionism. That is, at least with respect to the examples discussed, there is no reason to question the hegemony of causal accounts.
James Woodward’s contribution displays monist tendencies, as he explores whether and to what extent his well-known version of the counterfactual theory of explanation can be extended from its original causal interpretation to certain cases of non-causal explanation. Woodward defends the claim that such an extension is possible in at least two cases: first, if the relevant explanatory counterfactuals do not have an interventionist interpretation, and, second, if the truth of the explanatory counterfactuals is
supported by conceptual and mathematical facts. Finally, he discusses the role of information about irrelevant factors in (non-causal) scientific explanations.
Part II consists of contributions discussing detailed case studies of non-causal explanations from specific scientific disciplines. The case studies under discussion range from neuroscience over earth science to physics. The ambition of these chapters is to analyse in detail what makes a specific kind of explanation from one particular discipline non-causal.
Alisa Bokulich analyses a non-causal explanation from the earth sciences, more specifically from aeolian geomorphology (the study of landscapes that are shaped predominantly by the wind). Her case study consists in an explanation of regular patterns in the formation of sand ripples and dunes in deserts of different regions of earth and other planets. Bokulich uses this case study to argue for the “common core conception of non-causal explanation” in order to sharpen the concept of the non-causal character of an explanation. Moreover, she emphasizes that if one has a non-causal explanation for a phenomenon this does not exclude that there is also a causal explanation of the same explanandum.
Mazviita Chirimuuta focuses on a case study from neuroscience, efficient coding explanation. According to Chirimuuta, one ought to distinguish four types of explanations in neuroscience: (a) aetiological explanations, (b) mechanistic explanations, (c) non-causal mathematical explanations, and (d) efficient coding explanations. Chirimuuta argues that efficient coding explanations are distinct from the types (a)–(c) and are an often overlooked kind of explanation whose explanatory resources hinge on the implementation of an abstract coding scheme or algorithm. Chirimuuta explores ways in which efficient coding explanations go ‘beyond causation’ in that they differ from mechanistic and, more broadly, causal explanations. The global outlook of Chirimuuta’s chapter is monist in its spirit, as she indicates that all four types of explanations— including efficient coding explanations—answer what-if-things-had-been-different questions which are at the heart of counterfactual theories.
Steven French and Juha Saatsi investigate explanations from physics that turn on symmetries. They argue that a counterfactual-dependence account, in the spirit of Woodward, naturally accommodates various symmetry explanations, turning on either discrete symmetries (e.g., permutation invariance in quantum physics), or continuous symmetries (supporting the use of Noether’s theorem). The modal terms in which French and Saatsi account for these symmetry explanations throw light on the debate regarding the explanatory status of the Pauli exclusion principle, for example, and opposes recent analyses of explanations involving Noether’s theorem.
Margaret Morrison provides a rigorous analysis of the non-causal character of renormalization group explanations of universality in statistical mechanics. Morrison argues that these explanations exemplify structural explanations, involving a particular kind of transformation and the determination of ‘fixed points’ of these transformations. Moreover, Morrison discusses how renormalization group explanations exhibit important differences to other statistical explanations in the context of statistical mechanics
that operate by “averaging over microphysical details”. Although Morrison does not address the issue explicitly, it is clear that she rejects causal reductionism, and it is plausible to say that her non-causal characterization of renormalization group explanations is compatible with pluralism and monism.
Part III extends the analysis of non-causal explanations from the natural and social sciences to extra-scientific explanations. More precisely, the contributions in this part discuss explanatory proofs in pure mathematics and grounding explanations in metaphysics.
Mark Colyvan, John Cusbert, and Kelvin McQueen provide a theory of explanatory proofs in pure mathematics (aka intra-mathematical explanations). An explanatory proof does not merely show that a theorem is true but also why it is true. Colyvan, Cusbert, and McQueen pose the question whether explanatory proofs all share some common feature that renders them explanatory. According to their view, there is no single feature that makes proofs explanatory. Rather one finds at least two types of explanation at work in mathematics: constructive proofs (whose explanatory power hinges on dependence relations) and abstract proofs (whose explanatory character consists in their unifying power). Constructive and abstract proofs are two distinct ‘flavours’ of explanation in pure mathematics requiring different philosophical treatment. In other words, Colyvan, Cusbert, and McQueen make the case for explanatory pluralism in the domain of pure mathematics.
Lina Jansson analyses non-causal grounding explanations in metaphysics. In the flourishing literature on grounding, there is large agreement that grounding relations are explanatory and that they are explanatory in a non-causal way. But what makes grounding relations explanatory? According to some recent ‘interventionist’ approaches, the answer to this question should begin by assuming that grounding is a relation that is closely related to causation and, more precisely, that grounding explanations should be given an account in broadly interventionist terms (relying on structural equations and directed graphs functioning as representations of grounding relations). If these interventionist approaches were successful, they would provide a unified monist framework for ordinary causal and grounding explanations. However, Jansson argues that interventionist approaches to grounding explanations fail because causal explanations and grounding explanations differ with respect to the aptness of the causal models and grounding models underlying the explanations.
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